Let $x_{1}, …, x_{n}$ be several independent variables. Let $u = u(x_{1}, …, x_{n})$ be a function. Let $L(x_{1}, …, x_{n},u,\partial_{1}u,…,\partial_{n}u)$ be the Lagrangian whose action we want to extremize.
Euler Lagrange equation is stated as
$$\frac{\partial L}{\partial u} ~=~ \sum_{i=1}^{n}\frac{\partial}{\partial x_{i}}\frac{\partial L}{\partial (\partial_{i}u)} .$$
The problem is that the partial derivative with respect to $x_i$ is not really a partial derivative, rather it includes both implicit and explicit dependence on $x_i$. The notation is really unclear on this issue. Is there a way of indicating this dependency in the notation? Or is there another way of writing Euler Lagrange Equation in which this issue is bypassed altogether?
Best Answer
Yes, there exist a notation: Use total derivatives $$\frac{\mathrm{d}}{\mathrm{d} x^i} ~=~ \frac{\partial}{\partial x^i} ~+~ \partial_iu \frac{\partial }{\partial u} ~+~ \sum_j\partial_i\partial_ju \frac{\partial }{\partial (\partial_ju)} ~+~ \sum_{j\leq k}\partial_i\partial_j\partial_ku \frac{\partial }{\partial (\partial_j\partial_ku)} ~+~\ldots$$ so that the EL eqs. read $$0~=~\frac{\partial L}{\partial u} ~-~ \sum_i\frac{\mathrm{d}}{\mathrm{d} x^i}\frac{\partial L}{\partial (\partial_iu)} ~+~ \sum_{i\leq j}\frac{\mathrm{d}}{\mathrm{d} x^i}\frac{\mathrm{d}}{\mathrm{d} x^j}\frac{\partial L}{\partial (\partial_i\partial_ju)} ~-~\ldots. $$ Here the ellipsis "$\ldots$" denote possibly higher-order derivative terms in case the Lagrangian $L$ depends on higher-order derivatives of $u$.